Hello,
I have to find out if this series converges:
$\displaystyle \sum_{n=1}^{\infty}{\frac{1}{n}\left(e^\frac{1}{n}-1\right)}$
Any help would be appreciated.
$\displaystyle e^x=1+x+O(x^2)$
So $\displaystyle \frac{1}{n}\left(e^{\frac 1n}-1\right)=\frac{1}{n}\left(1+\frac{1}{n}+O\left(\fr ac{1}{n^2}\right)-1\right)=\frac{1}{n^2}+O\left(\frac{1}{n^2}\right)$.
Now $\displaystyle \sum\left(\frac{1}{n^2}+O\left(\frac{1}{n^2}\right )\right)<\infty$, from which you can conclude that the original series also converged.